Unbounded derivations tangential to compact groups of automorphisms,II |
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Authors: | Ola Bratteli Frederick M Goodman Palle ET Jørgensen |
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Institution: | 1. Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA;2. Department of Mathematics, University of Ottawa, Ottawa, Ontario, Canada |
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Abstract: | Let G be a compact abelian group, and τ an action of G on a C1-algebra , such that τ(γ)τ(γ)1 = τ(0) τ for all , where τ(γ) is the spectral subspace of corresponding to the character γ on G. Derivations δ which are defined on the algebra F of G-finite elements are considered. In the special case δ¦τ = 0 these derivations are characterized by a cocycle on with values in the relative commutant of τ in the multiplier algebra of , and these derivations are inner if and only if the cocycles are coboundaries and bounded if and only if the cocycles are bounded. Under various restrictions on G and τ properties of the cocycle are deduced which again give characterizations of δ in terms of decompositions into generators of one-parameter subgroups of τ(G) and approximately inner derivations. Finally, a perturbation technique is devised to reduce the case δ(F) ? F to the case δ(F) ? F and δ¦τ = 0. This is used to show that any derivation δ with D(δ) = F is wellbehaved and, if furthermore G = T1 and δ(F) ? F the closure of δ generates a one-parameter group of 1-automorphisms of . In the case G = Td, d = 2, 3,… (finite), and δ(F) ? F it is shown that δ extends to a generator of a group of 1-automorphisms of the σ-weak closure of in any G-covariant representation. |
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